Periodic solutions of semilinear impulsive periodic system on Banach space Original Research Article

In this paper, a class of semilinear impulsive periodic systems on Banach space is considered. Using impulsive periodic evolution operator, the T0T0-periodic PCPC-mild solution is introduced and suitable Poincaré operator is constructed. Showing the compactness of the Poincaré operator and using a new generalized Gronwall inequality with mixed type integral operators, we utilize the Leray–Schauder fixed point theorem to prove the existence of the T0T0-periodic PCPC-mild solutions. Our method is much different from methods of other papers. An example illustrates the applicability of our results.